Abstract A step-2 backward differential formula (BDF) temporal discretization scheme is constructed for nonlinear reaction-diffusion equation and superconvergence results are studied by mixed finite element method (FEM) with the elements Q 11 and Q 01 × Q 10 unconditionally. In particular, we apply an artificial regularization term to guarantee the energy stability of the step-2 BDF scheme. Splitting technique is utilized to get rid of the ratio between the time step size τ and the subdivision parameter h. Temporal error estimates in H 2 -norm are derived by use of the function's monotonicity, which leads to the regularities of the solutions for the time-discrete equations. Spatial error estimates in L 2 -norm are deduced to bound the numerical solution in L ∞ -norm. Unconditional superconvergence estimates of u n in H 1 -norm and q → n = ∇ u n in ( L 2 ) 2 -norm with order O ( h 2 + τ 2 ) are obtained. The global superconvergent properties are deduced through above results. Two numerical examples testify the theoretical analysis.

中文翻译：

---

BDF混合有限元非线性反应扩散方程能量稳定格式的超收敛分析

摘要 针对非线性反应-扩散方程构造了第2步后向微分公式(BDF)时间离散化方案，并采用混合有限元法(FEM)无条件地研究了元素Q 11 和Q 01 × Q 10 的超收敛结果。特别是，我们应用了人工正则化项来保证 step-2 BDF 方案的能量稳定性。利用分裂技术去除时间步长τ和细分参数h之间的比值。H 2 -范数中的时间误差估计是通过使用函数的单调性导出的，这导致时间离散方程的解具有规律性。推导出 L 2 -范数中的空间误差估计以限制 L ∞ -范数中的数值解。获得在 H 1 -范数和 q → n = ∇ un in ( L 2 ) 2 -范数的无条件超收敛估计，阶数为 O ( h 2 + τ 2 )。通过以上结果推导出全局超收敛性质。两个数值例子证明了理论分析。